The distance \(d_H\)  has been proven to be a metric for the space of all compact subsets of \(X\)  [1]}.
Translating the Hausdorff concept from point clouds to dendrograms, we consider the same two point clouds in the metric space \((X,d)\) , \(C=\lbrace x_1,\ldots , x_n\rbrace \)  and \(C^{\prime }=\lbrace y_1,\ldots , y_m\rbrace \)  and we consider \(T_C\)  and \(T_{C^{\prime }}\)  the single linkage hierarchical clustering dendrograms obtained from \(C\)  and \(C^{\prime }\)  respectively.
In the following, we prove the following result:
